Ancient Egyptian multiplication example

 

Ancient Egyptian and Russian Peasant

Russian peasant and Ancient Egyptian multiplication are both based on the principles of doubling and halving. They all work with powers of two, which makes the process simpler and more efficient. 

Revised: 

Russian Peasant Method:

Doubling and Halving: This method involves writing two numbers to be multiplied at the top of two columns. One column is repeatedly halved, ignoring any fractions, and the other is doubled.

Eliminating Even Numbers: Any row where the halved number is even is eliminated.

Summation: The remaining numbers in the doubled column are summed up to get the final product.

Example:

Multiplying 9 by 13:

Halve 9 and double 13: 4 (ignore the fraction) and 26.

Halve 4 and double 26: 2 and 52.

Halve 2 and double 52: 1 and 104.

Eliminate the rows with even numbers in the halved column (4 and 2).

Add the remaining numbers in the doubled column: 13 + 104 = 117.

Ancient Egyptian Method:

Doubling and Representation: Similar to the Russian method, it involves doubling one of the numbers. However, the other number is represented in terms of powers of 2.

Selecting Representations: The method identifies which powers of 2 can be combined to represent the second number.

Summation: The corresponding values from the doubled column are added to get the product.

Example:

Multiplying 9 by 13:

Write down 1 and double it to get 2, 4, 8 (stop at or before reaching the multiplier, which is 9).

Double 13 alongside: 26, 52, 104.

Since 9 can be represented as 8 + 1, select the rows with 8 and 1.

Add the corresponding numbers in the doubled column: 104 + 13 = 117.

Similarities and Differences:

Similarity: Both methods are based on the principle of doubling and halving. They utilize the decomposition of numbers into sums of powers of 2.

Difference: The Russian peasant method involves halving one number and eliminating rows based on even numbers, while the Ancient Egyptian method focuses on representing one number as a sum of powers of 2 and selecting the corresponding rows for summation.

Equivalence: They are equivalent because both methods essentially break down the multiplication process into additions of numbers that are powers of 2. This is a fundamental principle in binary arithmetic, where numbers are represented as sums of powers of 2.

In summary, while both methods use the concept of doubling and powers of 2, the Russian peasant method is more about halving and eliminating even rows, whereas the Ancient Egyptian method is about representing a number with powers of 2 and selecting the corresponding rows for summation. 

Babylonian word problems

The Babylonian word problems are practical in nature. They deal with real world problems. These problems were deeply rooted in the daily life and social needs of the Babylonians. While the Babylonians focused on solving specific problems, they began to develop general methods for approaching similar types of problems. They worked on algebraic problems without the symbolic notation or abstract symbols that we use today.

Babylonian Algebra

The Babylonians did not use abstract variables like we do today. Instead, they asked questions about specific quantities, such as length, area, or volume. Modern mathematical symbols were not used, and their notation was more like written language. 

The article emphasizes that the Babylonians were able to solve complex mathematical problems without using the abstractions we use today. Their approaches were practical and based on actual issues, indicating that while abstraction helps modern mathematics, it is not the only strategy for solving mathematical problems.

 

For number theory, we can explore properties of numbers by listing the numbers and finding the patterns and sequences.

Revised:

Babylonian Algebra:

Babylonian mathematics was highly practical, focusing on specific problems rather than general formulas. They dealt with concrete quantities like lengths, areas, and volumes directly.

Problems were often stated as word problems. They used tables of values to aid in calculations, such as tables for multiplication.

Instead of variables like x and y, Babylonians described problems using everyday language, referring to specific quantities and objects.

Many algebraic problems were solved geometrically. For example, solving a quadratic equation was approached by manipulating areas and lengths in a geometric context.

General Mathematical Principles Before Algebra

Before the development of algebra, many mathematical concepts were expressed through geometry. For example, the Pythagoreans used geometric diagrams to represent and explore numerical relationships.

Tables were a common tool for calculation. For instance, multiplication tables helped in performing complex calculations without the need for general formulas.

Problems and their solutions were often described verbally, using everyday language. This method was more narrative and less symbolic.

Mathematical principles were often taught and understood through practical examples and real-world applications, rather than abstract theory.

Stating General or Abstract Relationships Without Algebra: 

Number Theory: Instead of using variables and equations, patterns and properties of numbers could be explored through listing and visually examining sequences of numbers, much like the way prime numbers or perfect numbers were identified.

Geometry: Geometric shapes and figures can represent mathematical concepts. For example, the concept of the Pythagorean theorem can be demonstrated using a diagram of a right triangle with squares on its sides.

Calculus: The fundamental ideas of calculus, like limits and rates of change, could be explored through geometric slopes and areas under curves.

In summary, while modern algebra provides a powerful and efficient language for expressing and solving a wide range of mathematical problems, historical approaches demonstrate that it's possible to explore and convey complex mathematical ideas through more concrete and practical means. The Babylonians, for instance, effectively handled what we now recognize as algebraic problems, but they did so using specific, real-world scenarios and a more narrative style, without the abstract symbolism that characterizes modern algebra.

Babylonian-style base 60 multiplication table for the number 45

 

Why base 60?

Thinking of the Babylonians had base 60 system, the first reason I can come up with is that they might find base 10 system too small and have fewer factors than 60. 60s are commonly used in our daily life. For example, an hour has 60 minutes, and a minute has 60 seconds. This hour system has been used for a long time. Angles are also related to 60. A circle is divided into 360 degrees, which is 6 times 60. The longitude of the earth is also divided into 360 degrees, which is also related to 60. China has 12 animals and 5 elements for the zodiac. 12 times 5 is 60. This could also be the relationship with the base 60 system. 

Research shows that my guess is somehow correct. More factors for 60 can have a lot of flexibility for fractions and can make calculations easier. Also, finger-joint counting allows people to count to 12 with a single hand. This means that five repeated hand counts give 60. 

Crest of the Peacock

The first thing that surprises me about this chapter is that we used to accept only the mathematical tradition from Europe. Contributions from colonized peoples were devalued or ignored. I never noticed this, and I did not remember that most of the mathematical contributions I learned came from Europe. I was also surprised that this bias is not only limited to math and science related subjects, but also to social geography subjects. I feel sad about this, and I'm so glad that we are now acknowledging everyone's contribution. The last thing I am interested in is the Mayan understanding and use of zero in their number system. They have a lot of surprising areas outside of mathematics. It is important to know every culture and every contribution by reading the history. 

Why teach math history?

Before reading the article, I agree with the inclusion of mathematics history in mathematics education. I believe that incorporating math history will enrich students' understanding. My personal ideas about the benefits of teaching mathematics history are: 1. Helping students understand how mathematical concepts and formulas were developed and how they are used. 2. Students will be inspired by the achievements of mathematicians and will pursue mathematics in the future. I would like to incorporate math history into my own math teaching by giving students some historical math problems to solve and engaging students in proofs to understand how mathematicians of the past thought.

This article mentioned various ways of introducing the history and origin of the Pythagorean Theorem. This stopped me and reminded me of what I had learned about this theorem. It also summarized some of the challenges of integrating history into the math classroom. I agree with the idea about the potential for misconceptions. This is the key to ensuring that students receive accurate information about the history of mathematics. I am still in favor of incorporating the history of mathematics into mathematics education. But I noticed some more ways to do it from the article. And I would like to try them in the future.